Regarding the shaving of C1, I found this study that details 10 patients who had both Styloid removal and C1 shaving/tubercula drilling. 9 patients felt better after the operation or experienced resolution of their symptoms with one patient experiencing transient swallowing difficulty with facial drooping. I am not saying C1 shaving is free of complications but this study appears to be promising in terms of the safety of it. I know many doctors do not do for the fear of complications but I have seen many folks in here not improve after styloid removal only.
It is also important to note that one of the writers is part of the doctors list here and his name is Dr. Peter Nakaji. The average hospital stay was 2.1 days (range 1 - 6 days) and the average follow up was 11.8 months ( range 1 - 42 months) according to the study. See the study link below.
Styloidogenic Jugular Venous Compression Syndrome: Clinical Features and Case Series